Summary: We propose a notion of a quantum universal enveloping algebra for any Lie algebra defined by generators and relations which is based on the quantum Lie operation concept [V. K. Kharchenko, J. Algebra 217, 188–228 (1999; Zbl 0943.17016)]. This enveloping algebra has a PBW basis that admits a monomial crystallization by means of the Kashiwara idea. We describe all skew primitive elements of the quantum universal enveloping algebras for the classical nilpotent algebras of the infinite series defined by the Serre relations and prove that the above set of PBW-generators for each of these enveloping algebras coincides with the Lalonde-Ram basis of the ground Lie algebra with a skew commutator in place of the Lie operation. The similar statement is valid for Hall-Shirshov basis of any Lie algebra defined by one relation, but it is not so in the geneal case.